Local discontinuous Galerkin FEM for convex minimization
This work addresses a bottleneck in numerical analysis for degenerated convex minimization problems, offering incremental improvements in error control and adaptive algorithms.
The paper tackles the problem of suboptimal convergence rates in non-conforming finite element discretizations for convex minimization problems, achieving improved a priori convergence rates for energy errors and providing an a posteriori error control that enables adaptive mesh-refining with better performance than uniform methods.
The heart of the a priori and a posteriori error control in convex minimization problems is the sharp control of the approximation of the respective discrete and exact minimal energies. Conforming finite element discretizations for p-Laplace type minimization problems provide upper bounds of the energy difference with optimal convergence rates. Proven convergence rates for higher-order non-conforming finite element discretizations for the same problem class, however, are exclusively suboptimal. Thus the popular a posteriori error control within the two-energy principle, that generalize hyper-circle identities, appears unbalanced. The innovative point of departure in a refined analysis of two discontinuous Galerkin (dG) schemes exploits duality relations between a discrete primal and a semi-discrete dual problem. The infinite-dimensional dual problem leads to a tiny duality gap that even vanishes for polynomial low-order terms. For a class of degenerated convex minimization problems with two-sided $p$ growth, the novel duality provides improved a priori convergence rates for the error in the minimal energies. The motivating two-energy principle and some post-processing for a Raviart-Thomas dual variable provides an a posteriori error control, that also may drive adaptive mesh-refining. Computational benchmarks provide striking numerical evidence for improved convergence rates of the adaptive beyond uniform mesh-refining.